## Cooperative and Noncooperative Multi-Level ProgrammingTo derive rational and convincible solutions to practical decision making problems in complex and hierarchical human organizations, the decision making problems are formulated as relevant mathematical programming problems which are solved by developing optimization techniques so as to exploit characteristics or structural features of the formulated problems. In particular, for resolving con?ict in decision making in hierarchical managerial or public organizations, the multi level formula tion of the mathematical programming problems has been often employed together with the solution concept of Stackelberg equilibrium. However,weconceivethatapairoftheconventionalformulationandthesolution concept is not always suf?cient to cope with a large variety of decision making situations in actual hierarchical organizations. The following issues should be taken into consideration in expression and formulation of decision making problems. Informulationofmathematicalprogrammingproblems,itistacitlysupposedthat decisions are made by a single person while game theory deals with economic be havior of multiple decision makers with fully rational judgment. Because two level mathematical programming problems are interpreted as static Stackelberg games, multi level mathematical programming is relevant to noncooperative game theory; in conventional multi level mathematical programming models employing the so lution concept of Stackelberg equilibrium, it is assumed that there is no communi cation among decision makers, or they do not make any binding agreement even if there exists such communication. However, for decision making problems in such as decentralized large ?rms with divisional independence, it is quite natural to sup pose that there exists communication and some cooperative relationship among the decision makers. |

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### Contents

1 | |

6 | |

Optimization Concepts and Computational Methods | 10 |

22 Multiobjective programming | 13 |

222 Interactive multiobjective programming | 14 |

223 Fuzzy multiobjective programming | 16 |

23 Stochastic programming | 17 |

24 Genetic algorithms | 20 |

442 Numerical example | 119 |

45 Fuzzy decentralized twolevel linear programming | 121 |

451 Interactive fuzzy programming | 122 |

452 Numerical example | 127 |

46 Fuzzy twolevel linear 01 programming | 132 |

461 Interactive fuzzy programming | 133 |

462 Genetic algorithm with double strings | 134 |

463 Numerical example | 137 |

Noncooperative Decision Making in Hierarchical Organizations | 25 |

32 Twolevel linear programming | 31 |

322 Computational methods based on genetic algorithms | 33 |

323 Computational Experiments | 36 |

33 Twolevel mixed zeroone programming | 38 |

331 Facility location and transportation problem | 39 |

332 Computational methods based on genetic algorithms | 42 |

333 Computational Experiments | 47 |

34 Twolevel linear integer programming | 50 |

341 Computational methods based on genetic algorithms | 52 |

342 Computational Experiments | 56 |

35 Multiobjective twolevel linear programming | 59 |

351 Computational methods | 61 |

352 Numerical examples | 71 |

36 Stochastic twolevel linear programming | 75 |

361 Stochastic twolevel linear programming models | 76 |

362 Computational method for Vmodel | 78 |

363 Numerical example | 80 |

Cooperative Decision Making in Hierarchical Organizations | 83 |

42 Fuzzy two and multilevel linear programming | 86 |

421 Interactive fuzzy programming for twolevel problem | 87 |

422 Numerical example for twolevel problem | 93 |

423 Interactive fuzzy programming for multilevel problem | 97 |

424 Numerical example for multilevel problem | 102 |

43 Fuzzy twolevel linear programming with fuzzy parameters | 106 |

432 Numerical example | 111 |

44 Fuzzy twolevel linear fractional programming | 114 |

441 Interactive fuzzy programming | 115 |

47 Fuzzy twolevel nonlinear programming | 139 |

471 Interactive fuzzy programming | 141 |

Revised GENOCOP III | 146 |

473 Numerical example | 150 |

48 Fuzzy multiobjective twolevel linear programming | 153 |

481 Interactive fuzzy programming | 154 |

482 Numerical example | 161 |

49 Fuzzy stochastic twolevel linear programming | 166 |

491 Stochastic twolevel linear programming models | 167 |

492 Interactive fuzzy programming | 169 |

493 Numerical example | 171 |

494 Alternative stochastic models | 176 |

Some applications | 181 |

511 Problem formulation | 183 |

512 Maximization of proﬁt | 185 |

513 Maximization of profitability | 193 |

514 Discussions and implementation | 200 |

52 Decentralized twolevel transportation problem | 201 |

521 Problem formulation | 202 |

522 Interactive fuzzy programming | 207 |

53 Twolevel purchase problem for food retailing | 223 |

531 Problem formulation | 224 |

532 Parameter setting and Stackelberg solution | 226 |

533 Sensitivity analysis | 231 |

534 Multistore operation problem | 234 |

239 | |

248 | |

### Other editions - View all

Cooperative and Noncooperative Multi-Level Programming Masatoshi Sakawa,Ichiro Nishizaki No preview available - 2009 |

Cooperative and Noncooperative Multi-Level Programming Masatoshi Sakawa,Ichiro Nishizaki No preview available - 2011 |

### Common terms and phrases

ˆδ aspiration levels assigned drivers Level Bard coefﬁcients column vector Company L1 constraints crossover decentralized two-level decision makers decision making problems deﬁned denote DM1 and DM2 double strings efﬁciently feasible region ﬁnd ﬁrst phase ﬁtness value following problem food retailer formulated fuzzy goals fuzzy numbers fuzzy parameters fuzzy programming method genetic algorithms GENOCOP housing material manufacturer individual interactive fuzzy programming Iteration leader level DM linear fractional programming linear membership function linear programming problem lower level company maximin minimal satisfactory level multi-level n2-dimensional Nishizaki noncooperative nonlinear programming objective function value obtaining Stackelberg solutions orders to Company Pareto optimal purchase volume random variable ratio of satisfactory rational responses reference point roulette wheel selection row vector Sakawa satisfactory degrees satisﬁed shown in Table solution to problem speciﬁed subject to A1x1 subjectto termination conditions transportation two-level linear programming two-level programming problems upper bound upper level company V-model volume at city